# Department Seminars & Colloquia

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We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices D≥3; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of $\dot{H}^1$-bounded radial solutions to energy-critical heat equations in dimensions N≥7, building upon soliton resolution for such solutions. This is a joint work with Frank Merle (IHES and CY Cergy-Paris University).

We consider Calogero—Moser derivative NLS (CM-DNLS) equation which can be seen as a continuum version of completely integrable Calogero—Moser many-body systems in classical mechanics. Soliton resolution refers to the phenomenon where solutions asymptotically decompose into a sum of solitons and a dispersive radiation term as time progresses. Our work proves soliton resolution for both finite-time blow-up and global solutions without radial symmetry or size constraints. Although the equation exhibits integrability, our proof does not depend on this property, potentially providing insights applicable to other non-integrable models. This research is based on the joint work with Soonsik Kwon (KAIST).

In this lecture, we will discuss the formal/mathematical connection between the Boltzmann equation and the compressible Euler equation. For the mathematical justification we study the convergence of real analytic solutions of Boltzmann equations toward smooth solutions of the compressible Euler equation (before shock). This lecture will be accessible to graduate students.

In general relativity, spacetime is described by a (1+3)-dimensional Lorentzian manifold satisfying the Einstein equations, and initial data sets (i.e., fixed-time configurations) correspond to embedded spacelike hypersurfaces. The initial data sets are required to satisfy underdetermined PDEs called constraint equations -- in the language of differential geometry, these are exactly the Gauss and Codazzi equations. The goal of my talk will be to elucidate the flexibility of these objects -- specific results to be presented include extension, gluing, asymptotics-prescription, and parametrization of asymptotically flat initial data sets, often with sharp assumptions. Basic to our approach is a novel way to construct solution operators for divergence-type equations with prescribed support properties, which should be of independent interest. This part is based on joint work with Phil Isett (Caltech), Yuchen Mao (UC Berkeley), and Zhongkai Tao (UC Berkeley).

In this talk, we will discuss cylindrical and hypoelliptic extensions of Hardy, Sobolev, Rellich, Caffarelli-Kohn-Nirenberg, and other related functional inequalities. We will then concentrate on discussing their best constants, ground states for higher-order hypoelliptic Schrödinger-type equations, and solutions to the corresponding variational problems.

The analysis on the limiting behavior of solution is pivotal for equations in geometric analysis, mathematical physics and application in optimization. In 80s, Rene Thom conjectured that if an analytic gradient flow has a limit, then it approaches to the limit along a unique asymptotic direction. This represents a next-order question following the seminal works by Lojasiewicz and L. Simon. In 2000, Thom's conjecture was affirmatively proved by Kurdyka, Mostowski, and Parusinski for finite dimensional gradient flows. In this first part, we will discuss about the basics about theory of Lojasiewicz concerning the uniqueness of limits. Then we explore vast applications in PDEs which were initiated by Leon Simon.

Following the brief introduction to Lojasiewicz's theory in the first part, in the second part we discuss Thom's gradient conjecture and our recent joint work with Pei-Ken Hung where we generalized this conjecture to the class of PDEs. The result classifies the next-order asymptotics by revealing both the rate and the direction of convergence to the limit. Finally we talk about possible future applications and working directions.